Thorie Financire Relation risque rentabilit attendue 1

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Théorie FinancièreRelation risque rentabilité
attendue (1)

• Professeur André Farber

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Introduction to risk

• Objectives for this session
• 1. Review the problem of the opportunity cost of
  capital
• 2. Analyze return statistics
• 3. Introduce the variance or standard deviation
  as a measure of risk for a portfolio
• 4. See how to calculate the discount rate for a
  project with risk equal to that of the market
• 5. Give a preview of the implications of
  diversification

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Setting the discount rate for a risky project

• Stockholders have a choice
• either they invest in real investment projects of
  companies
• or they invest in financial assets (securities)
  traded on the capital market
• The cost of capital is the opportunity cost of
  investing in real assets
• It is defined as the forgone expected return on
  the capital market with the same risk as the
  investment in a real asset

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Three key ideas

• 1. Returns are normally distributed random
  variables
• Markowitz 1952 portfolio theory, diversification
• 2. Stock prices are unpredictable (Efficient
  market hypothesis)
• Kendall 1953 Movements of stock prices are
  random
• 3. Capital Asset Pricing Model
• Sharpe 1964 Lintner 1965
• Expected returns are function of systematic risk

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Preview of what follow

• First, we will analyze past markets returns to
  obtain an estimate of the historical market risk
  premium (the excess return earned by investing in
  a risky asset as opposed to a risk-free asset)
• The next step will be to understand the
  implications of diversification.
• We will show that
• diversification enables an investor to eliminate
  part of the risk of a stock held individually
  (the unsystematic - or idiosyncratic risk).
• only the remaining risk (the systematic risk) has
  to be compensated by a higher expected return
• the systematic risk of a security is measured by
  its beta (?), a measure of the sensitivity of the
  actual return of a stock or a portfolio to the
  unanticipated return in the market portfolio
• the expected return on a security should be
  positively related to the security's beta

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Capital Asset Pricing Model
Expected return
RM
Rj
Risk free interest rate
ßj
1
Beta
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Normal distribution
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Returns

• The primitive objects that we will manipulate are
  percentage returns over a period of time
• The rate of return is a return per dollar (or ,
  DEM,...) invested in the asset, composed of
• a dividend yield
• a capital gain
• The period could be of any length one day, one
  month, one quarter, one year.

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Belgium - Monthly returns 1951 - 1999
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Standard Poor 500
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Microsoft
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Normal distribution illustrated
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Risk premium on a risky asset

• The excess return earned by investing in a risky
  asset as opposed to a risk-free asset
• U.S.Treasury bills, which are a short-term,
  default-free asset, will be used a the proxy for
  a risk-free asset.
• The ex post (after the fact) or realized risk
  premium is calculated by substracting the average
  risk-free return from the average risk return.
• Risk-free return return on 1-year Treasury
  bills
• Risk premium Average excess return on a risky
  asset

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Total returns US 1926-2002
Source Ross, Westerfield, Jaffee (2005) Table 9.2
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Market Risk Premium The Very Long Run
The equity premium puzzle
Source Ross, Westerfield, Jaffee (2005) Table
9A.1
Was the 20th century an anomaly?
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Portfolio selection
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Risk and expected returns for porfolios

• In order to better understand the driving force
  explaining the benefits from diversification, let
  us consider a portfolio of two stocks (A,B)
• Characteristics
• Expected returns
• Standard deviations
• Covariance
• Portfolio defined by fractions invested in each
  stock XA , XB XA XB 1
• Expected return on portfolio
• Variance of the portfolio's return

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Example

• Invest 100 m in two stocks
• A 60 m XA 0.6
• B 40 m XB 0.4
• Characteristics ( per year) A B
• Expected return 20 15
• Standard deviation 30 20
• Correlation 0.5
• Expected return 0.6 20 0.4 15 18
• Variance (0.6)²(.30)² (0.4)²(.20)²2(0.6)(0.4)
  (0.30)(0.20)(0.5)
• s²p 0.0532 ? Standard deviation 23.07
• Less than the average of individual standard
  deviations
• 0.6 x0.30 0.4 x 0.20 26

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Diversification effect

• Let us vary the correlation coefficient
• Correlationcoefficient Expected return
  Standard deviation
• -1 18 10.00
• -0.5 18 15.62
• 0 18 19.7
• 0.5 18 23.07
• 1 18 26.00
• Conclusion
• As long as the correlation coefficient is less
  than one, the standard deviation of a portfolio
  of two securities is less than the weighted
  average of the standard deviations of the
  individual securities

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The efficient set for two assets
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Combining the Riskless Asset and a single Risky
Asset

• Consider the following portfolio P
• Fraction invested
• in the riskless asset 1-x (40)
• in the risky asset x (60)
• Expected return on portfolio P
• Standard deviation of portfolio

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Relationship between expected return and risk

• Combining the expressions obtained for
• the expected return
• the standard deviation
• leads to

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Choosing between 2 risky assets Sharpe ratio

• Choose the asset with the highest ratio of excess
  expected return to risk
• Example RF 6
• Exp.Return Risk
• A 9 10
• B 15 20
• Asset Sharpe ratio
• A (9-6)/10 0.30
• B (15-6)/20 0.45

Expected return
B
A
Risk
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Risk aversion

• Risk aversion
• For a given risk, investor prefers more expected
  return
• For a given expected return, investor prefers
  less risk

Expected return
Indifference curve
Risk
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Utility function

• Mathematical representation of preferences
• a risk aversion coefficient
• u certainty equivalent risk-free rate
• Example a 2
• A 6 0 0.06
• B 10 10 0.08 0.10 - 2(0.10)²
• C 15 20 0.07 0.15 - 2(0.20)²
• B is preferred

Utility
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Optimal choice with a single risky asset

• Risk-free asset RF Proportion 1-x
• Risky portfolio S Proportion x
• Utility
• Optimum
• Solution
• Example a 2

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The efficient set for two assets